Theorem iuneq2d 3995

Description: Equality deduction for indexed union. (Contributed by Drahflow, 22-Oct-2015.)

Hypothesis

Ref	Expression
iuneq2d.2	|- (ph -> B = C)

Assertion

Ref	Expression
iuneq2d	|- (ph -> U_x e. A B = U_x e. A C)

Distinct variable groups:   ph,x   x,A

Allowed substitution hints:   B(x)   C(x)

Proof of Theorem iuneq2d

Step	Hyp	Ref	Expression
1		iuneq2d.2	. . 3 |- (ph -> B = C)
2	1	adantr 451	. 2 |- ((ph /\ x e. A) -> B = C)
3	2	iuneq2dv 3991	1 |- (ph -> U_x e. A B = U_x e. A C)

Syntax hints:   -> wi 4   = wceq 1642   e. wcel 1710  U_ciun 3970

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-iun 3972

This theorem is referenced by:  iununi  4051